They never said it like that. Bayohne specifically said that it doesn't work like in FF11.
The cause of most of the confusion comes from the ambiguity over what "attack power" means. To some people, "attack power" means the "attack" stat on your attributes page. To other people, "attack power" means how much damage you do to an enemy. Many people use it interchangeably, and that leads to confusion.
So, as clearly as possible:
STR does not affect the "attack" stat that you see on your attributes page. The only way to increase the "attack" stat is to wear gear that has +attack power on it.
Adding either STR or the "attack power" stat raises your overall damage output. Both stats raise minimum and maximum damage.
Every 1 point of STR increases your damage output more than 1 point of "+attack power". This is roughly a 2:3 ratio as pointed out by Carraway's testing.
A piece of STR materia is roughly equal to a piece of ATK materia of the same quality since you get more ATK stat in an ATK materia, so just use whatever you got. There are some differences, but that only matters to hardcore players so don't even bother.